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A022008 Initial members of prime sextuplets (p, p+4, p+6, p+10, p+12, p+16). 61
7, 97, 16057, 19417, 43777, 1091257, 1615837, 1954357, 2822707, 2839927, 3243337, 3400207, 6005887, 6503587, 7187767, 7641367, 8061997, 8741137, 10526557, 11086837, 11664547, 14520547, 14812867, 14834707, 14856757, 16025827, 16094707, 18916477, 19197247 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Without the initial 7, this gives primes at which difference pattern X42424Y (X and Y >= 8) occurs in A001223. - Labos Elemer

Subsequence of A022007. - Zak Seidov, Nov 01 2011

The primes in a sextuplet a(n), a(n)+4, a(n)+6, a(n)+10, a(n)+12, a(n)+16 are consecutive since a(n)+2, a(n)+8 and a(n)+14 cannot be prime (multiple of 3). - Jean-Christophe Hervé, Sep 27 2014

The prime sextuplets begining at a(n) give the highest concentration of primes that can occur on an interval of 17 integers (apart intervals begining at p <7). It is conjectured that there is an infinite number of such sextuplets. - Jean-Christophe Hervé, Sep 27 2014

For n > 1, the 3 odd integers preceding and the 3 odd integers following the sextuplet are not prime : a(n)-2 = a(n)+18 = 0 (mod 5), a(n)-4 = a(n)+20 = 0 (mod 3), a(n)-6 = a(n)+22 = 0 (mod 7) and thus a(n) = 97 (mod 210 = 2*3*5*7). - Jean-Christophe Hervé, Sep 27 2014

All terms are congruent to 7 mod 30. - Zak Seidov, May 07 2017

LINKS

Zak Seidov, Table of n, a(n) for n = 1..1000

T. Forbes, Prime k-tuplets

R. J. Mathar, Table of Prime Gap Constellations

EXAMPLE

n=2 : 97, 101, 103, 107, 109, 113 are consecutive primes, while 91, 93, 95 and 115, 117 and 119 are not (cf. 4th comment about the border of composites).

MAPLE

for i from 1 to 2*10^5 do if [ithprime(i+1), ithprime(i+2), ithprime(i+3), ithprime(i+4), ithprime(i+5)] = [ithprime(i)+4, ithprime(i)+6, ithprime(i)+10, ithprime(i)+12, ithprime(i)+16] then print(ithprime(i)); fi; od; # Muniru A Asiru, Sep 03 2017

MATHEMATICA

lst = {}; Do[p = Prime[n]; If[PrimeQ[p+4] && PrimeQ[p+6] && PrimeQ[p+10] && PrimeQ[p+12] && PrimeQ[p+16], AppendTo[lst, p]], {n, 1000000}]; lst

Transpose[Select[Partition[Prime[Range[10^6]], 6, 1], Differences[#]=={4, 2, 4, 2, 4}&]][[1]] (* Harvey P. Dale, Mar 15 2015 *)

PROG

(PARI) p=2; q=3; r=5; s=7; t=11; forprime(u=13, 1e9, if(u-p==16 && p%3==1, print1(p", ")); p=q; q=r; r=s; s=t; t=u) \\ Charles R Greathouse IV, Mar 29 2013

(MAGMA) [p: p in PrimesUpTo(2*10^7) | IsPrime(p+4) and IsPrime(p+6) and IsPrime(p+10)and IsPrime(p+12) and IsPrime(p+16)]; // Vincenzo Librandi, Aug 23 2015

(Perl) use ntheory ":all"; say for sieve_prime_cluster(1, 1e8, 4, 6, 10, 12, 16); # Dana Jacobsen, Sep 30 2015

(GAP)

P:=Filtered([1, 3..2*10^7+1], IsPrime);;  I:=[4, 2, 4, 2, 4];;

P1:=List([1..Length(P)-1], i->P[i+1]-P[i]);;

A022008:=List(Positions(List([1..Length(P)-Length(I)], i->[P1[i], P1[i+1], P1[i+2], P1[i+3], P1[i+4]]), I), j->P[j]); # Muniru A Asiru, Sep 03 2017

CROSSREFS

Cf. A022007.

Cf. A001223, A052162, A052163, A052164, A052165, A052166, A052167, A052168, A047078.

Sequence in context: A005014 A201063 A157035 * A200504 A267641 A267669

Adjacent sequences:  A022005 A022006 A022007 * A022009 A022010 A022011

KEYWORD

nonn,easy

AUTHOR

Warut Roonguthai

STATUS

approved

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Last modified June 17 14:09 EDT 2019. Contains 324185 sequences. (Running on oeis4.)